References of related arts are listed in the following:                [1] P. P. Acarnley and J. F. Watson, “Review of position-sensorless operation of brushless permanent-magnet machines,” IEEE Trans. Ind. Electron., vol. 53, no. 2, pp. 352-362, April 2006;        [2] S. Nakashima, Y. Inagaki, and I. Miki, “Sensorless initial rotor position estimation of surface permanent-magnet synchronous motor,” IEEE Trans. Ind. Applicat., vol. 36, no. 6, pp. 1598-1603, November/December 2000;        [3] W. J. Lee and S. Ki Sul, “A new starting method of BLDC Motors without position sensor,” IEEE Trans. Ind. Applicat., vol. 42, no. 6, pp. 1532-1538, November/December 2006;        [4] Y. C. Chang and Y. Y. Tzou, “A new sensorless starting method for brushless DC motors without reversing rotation,” IEEE PESC Conf., pp. 619-624, June 2007;        [5] U.S. Pat. No. 7,334,854 to Chang et al.;        [6] U.S. Pat. No. 6,172,498 to Schmidt et al.;        [7] U.S. Pat. No. 5,028,852 to Dunfield;        [8] U.S. Pat. No. 5,569,990 to Dunfield;        [9] U.S. Pat. No. 6,229,274 to Verremara et al.; and        [10]Y. S. Lai, F. S. Shyu, and S. S. Tseng, “New initial position detection technique for three-phase brushless DC motor without position and current sensors,” IEEE Trans. Ind. Applicat., vol. 39, no. 2, pp. 485-491, March/April 2003.        
As shown in FIG. 1, a conventional two-pole permanent magnet synchronous motor (PMSM) 10 includes a rotor 12 and a stator 14. Since a PMSM is constructed with stator windings and a fixed rotor field supplied by a permanent magnet, absolute rotor position information is required to exactly control the motor torque. Resolvers, encoders, and Hall sensors are usually utilized for sensing the rotor position. However, these sensors increase the machine size and cost of the motor drive, and reduce the system reliability. Therefore, many researches have been presented to achieve position and speed sensorless control for PMSMs, for example in [1]. However, most of these researches suffer the same difficulty in detecting the rotor position at standstill. If the initial rotor position at standstill cannot be exactly detected, the starting torque of the motor decreases, and temporarily reversed rotation may occur at starting, which is not allowed in some applications, such as hard disks.
Several methods for detecting the initial rotor position without alignment have been proposed [2]-[10]. The principle of initial angle detection is illustrated in FIG. 2. The stator inductance is a function of rotor flux and stator current, and the stator current will slightly increase or decrease the stator saturation according to the direction of the induced field. The stator inductance is indirectly detected by detecting the stator current under a voltage V. FIG. 3 is a diagram showing a PMSM 20 driven by a pulse width modulation (PWM) inverter 22 conventionally. The PWM inverter 22 includes six power switches S1-S6, and by switching the six power switches S1-S6, different voltage vectors, such as twelve voltage vectors with thirty-degree resolution shown in FIG. 4, are provided for the PMSM 20. A shunt resistor Rdc is coupled with the PWM inverter 22, and a DC-link current idc flowing therethrough is a function of the motor current flowing through the motor 20. Those methods for initial rotor position detection at standstill can be classified into three types, namely peak current measurement [2]-[6], rise time measurement [7]-[9], and terminal voltage detection [10].
The estimation methods based on the inductance variation due to the magnetic saturation effect have been presented by measuring the peak currents in [2]-[5]. FIG. 5 is a diagram showing the DC-link current response with different inductances. Referring to FIGS. 3 and 5, different voltage vectors are applied to the PMSM 20 with a preset voltage vector conduction time duration ts and the peak values of the phase or DC-link current idc are used to indicate the initial rotor position. The rise time of the current idc reflects the time constant of the windings, which is smaller for smaller inductance. In order to get the maximum peak-current difference, the optimal time duration ts of the applied voltage vector should be around the average time constant of the stator winding [3]. The lower the equivalent inductance of the PMSM 20 is, the greater the peak value of the DC-link current idc will be at the end of the time duration ts. Since the inductance in the windings is a function of rotor flux, the rotor position reflects the difference of time constant. Therefore, the relative position between the rotor magnet and the stator winding can be determined by the peak current response resulting from different voltage vectors. However, this duration may cause the rotor slightly rotate and may also cause over-current conditions. Besides, a high-resolution analog-to-digital converter (ADC) is required for evaluating the peak-current difference to achieve accurate initial angle detection. Different approaches for detecting the inductance variation without directly sensing the peak current have been presented in [6]-[10]. In [6], a method for measuring the rate of current change with respect to time for each current to determine rotor position angles at standstill is presented. However, an ADC is still required with this method, and the determination of current change rate may have large variations due to the noise issue.
The initial rotor position detection based on rise time measurement also determines the initial rotor position according to inductance variation, except that the peak current is not measured directly. Instead of measuring the peak value or the change rate of the currents, the sign of the time difference between the rise times of the currents is detected for determining rotor position angles [7]. Only a comparator is required to compare the current with a preset threshold value to control the pulse widths of the induced voltages. However, if the rotor is perpendicular to one of the testing voltage vectors, the sign of the time difference may be ambiguous. Besides, only two phases will be conducted at the same time with the six testing vectors. In order to solve the ambiguity issue of [7], a method with combined the sign of the rise time difference or its magnitude of the rise time is presented along with look-up tables to determine the rotor position, and all three phases will be conducted by tying the third phase to high or low to further resolve the ambiguity for rotor position detection with the method [8]. Instead of using the time difference of the rise times of the currents conducted by two opposite voltage vectors, the shortest time period of the rise times of all tested currents is used to indicate the rotor position in [9]. It should be noted that the presented methods in [6]-[9] all utilize the rise times of the conducted currents to determine the rotor position angle. The methods for initial rotor position detection via rise time measurement base their accuracy on timers used for counting rise time, and are therefore more accurate than those methods relying on measurement by ADCs. Moreover, according to such methods, rise time difference is related only to inductance variation and is irrelevant to inductance values.
The initial rotor position detection via terminal voltage detection determines angular rotor position according to a current free-wheeling period. Although neither current sensors nor ADCs are required with the method presented in [10], three comparators are necessary for comparing the scaled terminal voltage levels, such as the voltage levels at the terminals a, b, and c shown in FIG. 3. The accuracy of the rotor position detection may be reduced with scaling the terminal voltages due to the noise issue. Besides, only sixty-degree resolution can be obtained with this method.
In order to avoid mistakes when applying the methods for initial rotor position detection based on rise time measurement, it is necessary to wait until the motor current drops to zero before switching to a next voltage vector. Since the back-EMF voltage of the motor is zero at stand still, the DC-link current under a voltage vector applied to a motor can be derived as
                              idc          =                                    Vdc              Req                        ⁡                          [                              1                -                                  ⅇ                                                            -                                              Req                        L                                                              ⁢                    t                                                              ]                                      ,                            [                  Eq          ⁢                      -                    ⁢          1                ]            where Vdc is the DC-link voltage, Req is the equivalent resistance corresponding to the applied voltage vectors, and L is the equivalent stator inductance. If the voltage drop of the free-wheeling diode is neglected, by setting a threshold current Ith for limiting the DC-link current idc, it can be derived the rise time of the DC-link current idc from zero to Ith as
                              tr          =                                    -                              L                Req                                      ⁢                          ln              ⁡                              (                                  1                  -                                                            Req                      ×                      Ith                                        Vdc                                                  )                                                    ,                            [                  Eq          ⁢                      -                    ⁢          2                ]            and the fall time of the DC-link current idc from Ith to zero as
                    tf        =                              -                          L              Req                                ⁢                                    ln              ⁡                              (                                                      -                                          Vdc                      Req                                                                            Ith                    -                                          Vdc                      Req                                                                      )                                      .                                              [                  Eq          ⁢                      -                    ⁢          3                ]            From the equations Eq-2 and Eq-3, the rise time tr and the fall time tf are both proportional to the stator inductance L which varies with the rotor position angle due to the magnetic saturation effect. However, all the current methods for initial rotor position detection based on rise time measurement use single-ended amplifier circuits to detect the motor current, so that only the rise time, but not the fall time, of the motor current can be detected, and in consequence the time that the motor current drops to zero cannot be precisely determined. According to the equation Eq-3, the time that the motor current drops to zero may vary with different voltage vectors. Therefore, it is necessary to set a long delay time after the motor current rises to the threshold value so as to ensure that the motor current will be zero when switching between different voltage vectors. As a result, a long detection time is needed at motor startup.
Therefore, it is desired an apparatus and method for initial rotor position detection with a shorter detection time.